HP Prime and TI-84 Plus: Method of Equal Proportions: Number of U.S. Representatives
Introduction
The 2016 United
States Elections still fresh on the minds of some Americans, even with talk of
repealing the Electoral College. But how
does the Electoral College work?
Each of the 50 states
receives a set amount of electoral votes, which is based on the number of House
Representatives and Senate members. Each
state gets two Senators. The number of House Representatives is determined by
popular. Every ten years, specifically 1990,
2000, 2010, 2020, and so on, the United States takes nationwide census. That population becomes the basis of
determining the number of House of Representatives. Currently, there are 435 members of the
House.
After the
election, each state gives the set number of electoral votes to the candidate
who wins the popular vote in that state.
For example, California, where I’m from, has 55 electoral votes. If Candidate A wins by popular vote in
California, that candidate gets 55 votes.
The exceptions are Maine and Nebraska, which use a congressional
district method.
Note:
In the Electoral College, 3 additional votes goes to the District of
Columbia. However, since D.C. is not a
state, it won’t get seats in the House of Representatives.
The Method of Equal Proportions: Determining
the Number of Seats
In order to
represent the population as fair as possible, methods must be used to
distribute the number of seats. The
method currently used is the Method of Equal Proportions, which has been in use
since 1940.
The first step
is to give each state one seat at the House of Representatives. For the United States, after each state gets
one seat, there are 385 seats to assign.
The population
of the states are recalculated by the following formula:
A_n = P / √(m *
(m + 1))
where P = the
state population, and m = the next potential seat (so for example, if the state
currently has 5 seats, m = 6)
A recursive
method, the method used in the program EQPROP is used:
A_1 = P / √2
A_n+1 = A_n * √(n/(n+2))
where n = the
number the seats the state currently has
Example:
Small Hypothetical Nation
Suppose we have
a small hypothetical nation, called the Country of Celestia that consists of
five states. The government is similar
to the United States, having both a House and Senate. Each state has 2 Senators. The five states have the following population
for which 24 seats need to be distributed:
Virgo
|
148,000
|
Andromeda
|
107,500
|
Orion
|
95,500
|
Sagittarius
|
95,000
|
Pegasus
|
93,000
|
Total
|
539,000
|
As the first
step, each state gets 1 seat. That means
there are 19 seats remaining. (24 -
5) I am going to use the recursive
definition.
We need to
adjust the population (to determine the priority number) by dividing each
population by √2. Note: calculations in this example are rounded to
the nearest integer.
Virgo
|
104,652
|
1
|
Andromeda
|
76,014
|
1
|
Orion
|
67,259
|
1
|
Sagittarius
|
67,175
|
1
|
Pegasus
|
65,761
|
1
|
The next seat
(6 out of 24), will go to Virgo because Virgo has the highest adjusted population. Then Virgo’s next adjusted population will
be: 104,652 * √(1/3) = 60,421
Virgo
|
60,421
|
2
|
Andromeda
|
76,014
|
1
|
Orion
|
67,259
|
1
|
Sagittarius
|
67,175
|
1
|
Pegasus
|
65,761
|
1
|
Seat #7, goes
to Andromeda, because Andromeda now has the largest adjusted population at
76,014. Adjusting the population: 76,014 * √(1/3) = 43,887, and Andromeda gets
another seat.
After 7 seats,
this is what the adjusted population looks like:
Virgo
|
60,421
|
2
|
Andromeda
|
43,887
|
2
|
Orion
|
67,259
|
1
|
Sagittarius
|
67,175
|
1
|
Pegasus
|
65,761
|
1
|
We continue:
Virgo
|
Andromeda
|
Orion
|
Sagittarius
|
Pegasus
|
||||||
Seat #
|
Adj. Pop.
|
# Seats
|
Adj. Pop.
|
# Seats
|
Adj. Pop.
|
# Seats
|
Adj. Pop.
|
# Seats
|
Adj. Pop.
|
# Seats
|
8
|
60421
|
2
|
43887
|
2
|
38832
|
2
|
67175
|
1
|
65761
|
1
|
9
|
60421
|
2
|
43887
|
2
|
38832
|
2
|
38784
|
2
|
65761
|
1
|
10
|
60421
|
2
|
43887
|
2
|
38832
|
2
|
38784
|
2
|
37967
|
2
|
11
|
42724
|
3
|
43887
|
2
|
38832
|
2
|
38784
|
2
|
37967
|
2
|
12
|
42724
|
3
|
31033
|
3
|
38832
|
2
|
38784
|
2
|
37967
|
2
|
13
|
33094
|
4
|
31033
|
3
|
38832
|
2
|
38784
|
2
|
37967
|
2
|
14
|
33094
|
4
|
31033
|
3
|
27458
|
3
|
38784
|
2
|
37967
|
2
|
15
|
33094
|
4
|
31033
|
3
|
27458
|
3
|
27424
|
3
|
37967
|
2
|
16
|
33094
|
4
|
31033
|
3
|
27458
|
3
|
27424
|
3
|
26847
|
3
|
17
|
27021
|
5
|
31033
|
3
|
27458
|
3
|
27424
|
3
|
26847
|
3
|
18
|
27021
|
5
|
24038
|
4
|
27458
|
3
|
27424
|
3
|
26847
|
3
|
19
|
27021
|
5
|
24038
|
4
|
21269
|
4
|
27424
|
3
|
26847
|
3
|
20
|
27021
|
5
|
24038
|
4
|
21269
|
4
|
21243
|
4
|
26847
|
3
|
21
|
22837
|
6
|
24038
|
4
|
21269
|
4
|
21243
|
4
|
26847
|
3
|
22
|
22837
|
6
|
24038
|
4
|
21269
|
4
|
21243
|
4
|
20796
|
4
|
23
|
22837
|
6
|
19627
|
5
|
21269
|
4
|
21243
|
4
|
20796
|
4
|
24
|
19777
|
7
|
19627
|
5
|
21269
|
4
|
21243
|
4
|
20796
|
4
|
Final
Distribution for Celestia (our example):
Virgo, 7 House seats, Andromeda, 5 House seats, Orion, Sagittarius, and
Pegasus get 4 each.
On to the
programming!
The Program EQPROP
The program
EQPROP takes two arguments: the list of
populations, and the number of House of Representative seats to be
assigned. The program assumes that two
Senators will also be assigned.
Output: A matrix of three columns:
Column 1: The population of each state. The population is sorted in descending order.
Column 2: The number of House Representatives.
Column 3: The number of House Representatives plus the
2 senators.
For the TI-84
Plus: L1 is used as the population list,
lists L2, L3, and L4 are used for calculations, and the results are returned in
Matrix [A].
HP Program EQPROP
EXPORT EQPROP(lp,n)
BEGIN
// Method of equal proportions
// 2016-11-18 EWS
// population, no of seats
LOCAL la,s,lr,k,m,p,w;
// initialization
lp:=REVERSE(SORT(lp));
la:=lp/√2;
s:=n-SIZE(lp);
lr:=MAKELIST(1,X,1,SIZE(lp));
// loop
FOR k FROM s DOWNTO 1 STEP 1 DO
m:=MAX(la);
p:=POS(la,m);
w:=lr(p);
la(p):=la(p)*√(w/(w+2));
lr(p):=w+1;
END;
// output, organize matrix
// [population, senate, + house]
LOCAL l2,m2,l3;
l3:=lr+2;
l2:={lp,lr,l3};
l2:=TRN(list2mat(l2,3));
m2:=l2(1);
RETURN m2;
END;
TI-84 Plus Program EQPROP
"EQUAL
PROPORTIONS"
"2016-11-18
EWS"
Input "POP.
LIST: ",L₁
Input "NO. OF
SEATS: ",N
SortD(L₁)
L₁→L₂
N-dim(L₂)→S
L₁→L₃
Fill(1,L₃)
L₂/√(2)→L₂
"LOOP"
For(K,S,1,1)
max(L₂)→M
"POS"
1→P
While L₂(P)≠M
1+P→P
End
"REST"
L₃(P)→W
L₂(P)*√(W/(W+2))→L₂(P)
W+1→L₃(P)
End
"OUTPUT"
L₃+2→L₄
List>matr(L₁,L₃,L₄,[A])
Pause [A]
Sources:
Burnett,
Kristin D. “Congressional Apportionment: 2010 Census Briefs” U.S. Census Bureau. November 2011. Download the PDF file here: http://www.census.gov/library/publications/2011/dec/c2010br-08.html
Wikipedia “Electoral College (United States)” https://en.wikipedia.org/wiki/Electoral_College_(United_States) Retrieved November 16, 2016
Wikipedia “United
States Congressional Apportionment” https://en.wikipedia.org/wiki/United_States_congressional_apportionment#cite_note-22 Retrieved November 16, 2016
This blog is
property of Edward Shore, 2016